First-principles predictions of electronic properties of GaAs_1-x-yP_yBi_x and GaAs_1-x-yP_yBi_x-based heterojunctions

High-power infrared light-emitting diodes (LEDs) and laser diodes (LDs) have important applications in a number of areas, such as telecommunication, material processing, medical applications, and military defense. However, the efficiency of LEDs and LDs operating at high current densities (over ∼30 A/cm² for LEDs^1,2 or over ∼3 kA/cm² for LDs^3,4) dramatically decreases, a phenomenon known as efficiency droop.^5,6 To achieve high-power output, the usual remedy is employing parallel diode arrays operating around an optimal current density. This method, however, increases the volume, complexity, and cost of devices, and in turn prohibits the more widespread applications of high-power infrared LEDs and LDs.

The efficiency droop can be induced by three major sources: defects, electron leakage, and Auger recombination.^5–7 Recent experiments have confirmed that Auger recombination is the major cause of the efficiency droop in InGaN/GaN LEDs,⁸ a finding possibly true for other similar systems. Because direct Auger recombination is roughly proportional to $n3e−CEg/kBT$⁠,⁷ where n is carrier density, E_g the band gap, and C a material-dependent coefficient, this mechanism is especially significant for devices operating at high current density and with small band gap, as confirmed in a number of different materials.^5,9 For GaAs-based infrared diodes, it has been proposed^10,11 to both decrease the band gap and increase the spin-orbit (SO) splitting by Bi alloying to reduce one type of Auger recombination process, where a Conduction electron recombines with a Heavy hole and the released energy subsequently excites a Spin-split-off valence electron into a Heavy hole band (CHSH). A recent estimation¹² indicated that the rate of the CHSH process is 35% to 23% of the total Auger recombination rate, when the band gap of GaAs-based alloy changes from 1.4 to 0.7 eV, respectively.

However, it has been difficult to grow the ternary alloy GaAs_1–xBi_x with good structural^13,14 and optical¹⁵ properties because of the various defects in this alloy.^16,17 Recently, we fabricated a quaternary alloy GaAs_1–x–yP_yBi_x with P fraction up to 30.6% and Bi fraction up to 8.5% using metal-organic vapor phase epitaxy.¹⁸ Through co-incorporation of P and Bi into GaAs, it is feasible to enhance the solubility of Bi because of the relatively small atom size of P and meanwhile to tune the lattice and electronic properties. In this letter, we present our theoretical investigations of two essential issues associated with applying this alloy to the active layer of infrared LEDs and LDs. The first issue is the band gap and SO splitting of GaAs_1–x–yP_yBi_x, which determines the CHSH process of Auger recombination. The second issue is the band offsets of GaAs/GaAs_1–x–yP_yBi_x heterojunction, a property determining the carrier confinement in the double heterojunction GaAs/GaAs_1–x–yP_yBi_x/GaAs and subsequently its internal quantum efficiency.

We carry out all calculations using density functional theory (DFT) as implemented in the Vienna Ab initio Simulation Package.^19,20 The local density approximation (LDA) functional is employed, and SO coupling is included to describe the important special relativity effects in Bi atoms. The projector augmented wave method is used with the following potentials: Ga_GW (4s²4p¹), As_GW (4s²4p³), P_GW (3s²3p³), and Bi_d_GW (5d¹⁰6s²6p³). The plane-wave energy cutoff is set to 400 eV. The supercells in the band structure and band offset calculations consist of 3 × 3 × 3 and 2 × 2 × 6 of the GaAs conventional cells, including 216 and 192 atoms or a volume of ∼16.9 × 16.9 × 16.9 and ∼11.2 × 11.2 × 33.9 ų, respectively. In band offset calculations, a heterojunction is constructed along [001] and the two different materials have equal number of atoms. We predict a binding energy of 11 meV for P and Bi on neighboring As sites relative to isolated P and Bi dopants, suggesting that the P and Bi dopants form a nearly ideal solid solution at relevant synthesis temperatures, where the k_BT is much larger than 11 meV. Thus, we randomly distribute P and Bi atoms on the As sites using the special quasirandom structure (SQS) approach. The SQS with pair and triplet clusters within a radius of 5.8 Å is optimized with a Monte Carlo algorithm, as implemented in the ATAT code.²¹ The Monkhorst-Pack k-point grids for Brillouin zone sampling are 3 × 3 × 3 and 6 × 6 × 1 for the 216 and 192-atom structures, respectively. The lattice length is relaxed either in all directions for unstrained structures or only in the [001] direction for structures with (001) plane strained to GaAs(001). The remaining force on each atom is less than 0.01 eV/Å after relaxation.

The band gap underestimation by LDA functional is fixed by operating on the LDA gap with a linear function: 0.88 eV + 1.13 E^g, where E^g is the LDA band gap. This correction gives results in good agreement with previous experimental band gaps (see the supplementary material). Band offsets of the GaAs/GaAs_1–x–yP_yBi_x heterojunction are calculated based on positions of the valance band maximum (VBM) and conduction band minimum (CBM) relative to macroscopic electrostatic potential (MEP)²² with a procedure detailed in the supplementary material.

We investigate 25 compositions of GaAs_1–x–yP_yBi_x, with 0 ≤ x ≤ 0.10 and 0 ≤ y ≤ 0.40 (see supplementary material), and consider both fully relaxed structures and ones strained to GaAs(001). The DFT-based band gaps are fitted with Taylor series for easy calculations of other unexamined compositions, a method used also in the band gap study of GaAsNBi.²³ The fitting result of fully relaxed GaAs_1–x–yP_yBi_x is shown in Eq. (1)

The contour plots for the band gap of fully relaxed and strained structures are shown in Figs. 1(a) and 1(b), respectively. With increasing alloying content, P increases the band gap but Bi greatly reduces it. The relatively straight contour lines indicate that P and Bi atoms have weak cooperative effects to the band gap. For the strained case, the quaternary alloy possesses band gap less than that of GaAs (1.42 eV) under the condition y < 8.10 x – 37.55 x²+ 159.22 x³. The lattice-match condition is y = 4.35 x according to Vegard's law and the calculated lattice lengths of GaP (5.47 Å), GaAs (5.62 Å), and GaBi (6.27 Å) bulks. As we will show in the next paragraph, such Vegard's lattice-match condition is different from the one where the fully relaxed and strained structures have the same band gaps.

Figure 1(c) shows that the band gap difference between the strained and fully relaxed structures increases approximately linearly when the in-plane strain changes from tension to compression. Under the condition y = 2.26 x + 10.34 x²+ 9.91 x³ − 321.51 x⁴ (blue dashed line), the strained structures have the same band gaps as the fully-relaxed counterpart, which means that the former are lattice-matched with GaAs(001). Such lattice-match condition is close to the one where the lattice lengths of fully relaxed structures equate with those of GaAs (y = 2.66 x + 1.87 x²+ 0.91 x³), and is expected to be more accurate than the one based on Vegard's law, which is disobeyed in a number of materials.²⁴ We therefore consider the lattice-match line in Fig. 1(c) as a reasonable lattice-match condition. The band gap change by the strain effect is −0.21 to 0.05 eV in the examined composition range. Detailed analyses based on mechanical properties and DFT-based band gaps (see supplementary material) show that the strain effect on the band gap can be described by

where E_g and a are the band gap and lattice parameter of the fully relaxed structure, and ΔE_g and Δa are the band gap and in-plane lattice parameter changes of the strained alloy relative to its fully relaxed counterpart, respectively; the standard error of the coefficient is 0.24. With Eq. (3), the band gap change induced by the strain of other substrates can be easily evaluated.

To understand how the atomic radius and chemistry of P and Bi atoms influence the band gap change, we first examine the geometrical distortions of a heavily alloyed structure, GaAs_0.5P_0.398Bi_0.102. Figure 2(a) shows that local distortions are larger in the Ga layer than in the group-V layer, with an average displacement of 0.19 Å in the former relative to 0.09 Å in the latter. This phenomenon is related to the tetrahedral coordination in GaAs bulk: all defects on the As sites have locally centrosymmetric strain and thus stay near the As sites, but they push/pull four nearby Ga atoms. Statistics of the chemical bonds in Fig. 2(b) shows that the average lengths of the Ga-P, Ga-As, and Ga-Bi bonds are ∼2.37, ∼2.44, and ∼2.62 Å, respectively. Note that the Ga-As bond length is 2.43 Å in GaAs bulk. Therefore, the major structural changes can be summarized as follows: P atoms slightly pull Ga atoms close but Bi markedly pushes Ga atoms away, a phenomenon consistent with the order of covalent radii of P (1.07 Å), As (1.19 Å), and Bi (1.48 Å).²⁵ 

Then, we separately assess two influences associated with the P and Bi incorporation, the change of chemistry and the relaxation of structure, to understand the origin of band gap change induced by alloying. We choose GaAs_0.805P_0.093Bi_0.102 as an example, because it possesses band gap (0.93 eV) significantly different from that of GaAs bulk. First, substitution of all the P and Bi atoms in the fully relaxed GaAs_0.805P_0.093Bi_0.102 with As atoms generates a band gap of 1.08 eV, close to the 0.93 eV band gap of the optimized structure. Second, shift of the atom positions in fully relaxed GaAs_0.805P_0.093Bi_0.102 back to the reduced coordinates of GaAs bulk generates a band gap of 1.31 eV, very close to the 1.34 eV band gap of GaAs bulk. The two results indicate that the band gap changes in GaAs_1–x–yP_yBi_x are mainly caused by the structural changes induced by P and Bi atoms, and the chemical differences among As, P, and Bi atoms are not critical. This result is different from the foundation of the widely used valence band anti-crossing (VBAC) model in Bi-containing III–V alloys, where the model totally neglects structural distortions and ascribes the band gap changes to interactions between the valence bands of host semiconductor and the states of the alloying elements.²⁶ One prediction of the VBAC model is that the Bi-related states are localized near the valance-band edge.²⁶ This prediction, however, is in contradiction with our projected density of states results, which show that the Bi states are delocalized across the whole band structure, similar to the As states (see supplementary material). Therefore, a fitted VBAC model should be interpreted as an empirical way of describing primarily the effects of structural distortions.

Similar to the band gap of GaAs_1–x–yP_yBi_x, we also fit the DFT SO splitting of fully relaxed structures with a Taylor series as expressed in Eq. (4)

The contour plots for the SO splitting of fully relaxed and strained structures are shown in Figs. 3(a) and 3(b), respectively. As expected, P atoms decrease the SO splitting but Bi atoms increase it. Different from the strain effects on band gap, both strong tensile and compressive strains enhance the SO splitting (Fig. 3(c)), although the changes are less than ∼0.07 eV in the examined composition range. To obtain SO splitting greater than that of GaAs (0.35 eV), the composition of the strained alloy should satisfy y < 74.38 x. GaAs_1–x–yP_yBi_x with lower Auger recombination ratio than GaAs should possess a smaller band gap (y < 8.10 x − 37.55 x²+ 159.22 x³) and a larger SO splitting (y < 74.38 x) than GaAs, which therefore sets up a composition requirement of at least y < 8.10 x − 37.55 x²+ 159.22 x³.

To verify the abovementioned results, we compare our predictions of the band gap and SO splitting with available experiments. Our theoretical results are generally in good agreement with the experimental band gap²⁷ of GaAs_1–yP_y and band gap^11,28–30 and SO splitting^11,30,31 of GaAs_1–xBi_x (see Fig. S4 in supplementary material). The largest differences between the experimental data and Eqs. (1b), (1c), and (4b) are about 0.03, 0.02, and 0.08 eV, respectively. The agreement is also reasonably good with the limited experimental band gap¹⁸ of GaAs_1–x–yP_yBi_x. According to Eq. (2), the band gaps of strained structure with (x, y) = (0.031, 0.280), (0.070, 0.273), and (0.085, 0.230) are 1.47, 1.28, and 1.18 eV, which have an average deviation of about 0.08 eV relative to the experimental values of 1.39, 1.19, and 1.11 eV, respectively.

To examine if efficient carrier confinement can be realized in GaAs_1–x–yP_yBi_x-based LEDs and LDs, we investigate the band offsets of GaAs/GaAs_1–x–yP_yBi_x heterojunctions. Figure 4 compares the band diagrams of three heterojunctions with 8.3% Bi and three P contents: GaAs/GaAs_0.917Bi_0.083, GaAs/GaAs_0.771P_0.146Bi_0.083, and GaAs/GaAs_0.625P_0.292Bi_0.083, with respective lattice mismatch of 0.99%, 0.65%, and 0.31% based on our calculated lattice parameters of the fully relaxed GaAs and GaAs_1–x–yP_yBi_x. All the three quaternary alloys satisfy the composition requirement of having less Auger recombination than GaAs bulk (y < 8.10 x − 37.55 x²+ 159.22 x³). Figures 4(a) and 4(b) show that GaAs/GaAs_0.917Bi_0.083 and GaAs/GaAs_0.771P_0.146Bi_0.083 have the type-I (straddling gap) band diagram, which is favorable for optical transitions, and the VBM (CBM) band offsets are 240 (220) and 170 (140) meV for the former and latter heterojunctions, respectively. Because the band offsets of the two heterojunctions are 5.4–9.2 times of the thermal energy at room temperature (∼26 meV at 300 K), they are expected to exhibit effective carrier confinement and low leakage current at room temperature.³² However, Fig. 4(c) shows that GaAs/GaAsP_0.292Bi_0.083 heterojunction owns the type-II (staggered gap) band diagram with a VBM (CBM) band offset of 190 (20) meV, and thus is less useful for utilization in optical transitions than the previous two. These results demonstrate that by carefully choosing the composition of GaAs_1–x–yP_yBi_x, it is possible to obtain heterojunctions that simultaneously own large band offsets, small lattice mismatch, and band gap and SO splitting resulting in reduced Auger recombination.

It is worthwhile to point out that the previous estimation³³ of band offsets based on the band structures of the two bulk materials forming the heterojunction is likely unreliable for GaAs_1–x–yP_yBi_x. This is because the interface of a heterojunction usually induces electric dipoles, which lead to the MEP offset.³⁴ For the three examined heterojunctions, the MEP offsets are 90–240 meV, which are comparable to or even greater than the band offsets. The estimated band offsets based on bulk band structures, which neglect the interface dipole moment, therefore, possess significant deviations.

In summary, we calculated the band gap and SO splitting of GaAs_1–x–yP_yBi_x and the band offsets of GaAs/GaAs_1–x–yP_yBi_x heterojunctions using DFT. Analytic expressions of the band gap and SO splitting dependences on compositions were obtained, which are consistent with available experimental results. The band gap changes were found to be primarily contributed by the local distortions around P and Bi atoms. A composition requirement for GaAs_1–x–yP_yBi_x to achieve lower Auger recombination ratio than GaAs was obtained. Finally, we found that GaAs/GaAs_1–x–yP_yBi_x heterojunctions with achievable compositions and small lattice mismatch can exhibit the type-I band diagram and their band offsets can be tuned to suit the requirements of infrared LED and LD applications.

See supplementary material for the (a) procedure to obtain band offsets, (b) linear correction to DFT band gaps, (c) fitting errors of Eqs. (1), (2), (4), and (5), (d) reasoning of the strain effects on band gap, (e) projected density of states of GaAs_0.5P_0.398Bi_0.102, (f) comparison between Eqs. (1b), (1c), and (4b) with experimental results, and (g) complete list of the band gap, SO splitting, and lattice length.

This research was primarily supported by NSF through the University of Wisconsin Materials Research Science and Engineering Center (Grant No. DMR-1121288). The authors gratefully acknowledge the use of clusters supported by NSF through the University of Wisconsin Materials Research Science and Engineering Center (Grant No. DMR-1121288). Computing resources in this work also benefited from the use of the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant No. ACI-1053575 and the compute resources and assistance of the UW-Madison Center for High Throughput Computing (CHTC) in the Department of Computer Sciences.